In conventional textbooks
the basics of quantum theory
of unitary systems
are usually explained
in Schrödinger
representation alias 
Schrödinger picture (SP, [1]}).
The pure state
is represented there by a
time-dependent ket-vector element
of a Hilbert space
\({\cal L}\) . The evolution
is assumed generated by a self-adjoint Hamiltonian \(\mathfrak {h}\) .
The one-to-one correspondence between the
self-adjointness of \(\mathfrak {h}\)  and the unitarity of the
evolution was given the rigorous
mathematical form by Stone [2]}.
In this sense, the
Bender's and Boettcher's
claim [3]}
that the
unitary evolution
could be
also generated by a non-Hermitian
Hamiltonian \(H \ne H^\dagger \) 
sounded, initially, contradictory [4]}.
